An Integrated Electric Power Supply Chain and Fuel Market Network Framework: Theoretical Modeling with Empirical Analysis for New England

Transcription

1 An Integrated Electric Power Supply Chain and Fuel Market Network Framework: Theoretical Modeling with Empirical Analysis for New England Zugang Liu and Anna Nagurney Department of Finance and Operations Management Isenberg School of Management University of Massachusetts Amherst, Massachusetts August 2007; revised April 2008 Abstract: In this paper, we develop a novel electric power supply chain network model with fuel supply markets that captures both the economic network transactions in energy supply chains and the physical network transmission constraints in the electric power network. The theoretical derivation and analyses are done using the theory of variational inequalities. We then apply the model to a specific case, the New England electric power supply chain, consisting of 6 states, 5 fuel types, 82 power generators, with a total of 573 generating units, and 10 demand market regions. The empirical case study demonstrates that the regional electric power prices simulated by the proposed model very well match the actual electricity prices in New England. We also compute the electric power prices under natural gas and oil price variations. The empirical examples illustrate that both the generating unit responsiveness and the electric power market responsiveness are crucial to the full understanding and determination of the impact of the residual fuel oil price on the natural gas price. Finally, we utilize the model to quantitatively investigate how changes in the demand for electricity influence the electric power and the fuel markets from a regional perspective. The theoretical model can be applied to other regions and multiple electricity markets to quantify the interactions in electric power/energy supply chains and their effects on flows and prices in which deregulation is taking place. 1

2 1. Introduction Electric power systems provide a critical infrastructure for the functioning of our modern economies and societies. Electric power lights (and cools) our homes, our commercial and industrial enterprises, powers our computers, and enables the production and dissemination of goods and services worldwide. It is undeniably an essential form of energy, whose absence and/or unavailability, can have profound and lasting impacts in both the developed and developing corners of the globe. In order to understand the availability and, ultimately, reliability and vulnerability of electric power and the underlying systems, one must place the systems in the context of the industrial setting. For example, in the case of the United States, the electric power industry possesses more than half a trillion dollars of net assets, generates $220 billion annual sales, and consumes almost 40% of domestic primary energy [17, 18]. Currently, the electric power industry in the US is undergoing a deregulation process from once highly regulated, vertically integrated monopolistic utilities to emerging competitive markets [28, 32, 70]. This deregulation process has caused major changes to the electric power industry, and requires a deep and thorough identification of the structure of the emerging electricity supply chains, as well as new paradigms for the modeling, analysis, and computations for electric power markets. Smeers [66] reviewed a wide range of models of energy markets with various market power assumptions (see also, [1, 8, 9, 26, 48, 60, 65, 70, 71]). Hogan [31] proposed a market power model to study strategic interactions in an electricity transmission network. More recently, Chen and Hobbs [11] proposed an oligopolistic electricity market model with a nitrogen oxide permit market, and provided examples based on the PJM market (Pennsylvania, New Jersey, and Maryland). Nagurney and Matsypura [57], in turn, presented an electric power supply chain network model which provided an integrated perspective for electric power generation, supply, transmission, as well as consumption. Wu et al. [69] considered the generators generating unit portfolios and reformulated the electric power supply chain network model as a user-optimal transportation network model (see also [54]). Nagurney et al. [55] also established the connections between electric power supply chain networks and transportation networks, and developed a dynamic electric power supply chain network with time-varying 2

3 demands. This paper focuses on the relationship and interaction between electric power supply chains and other energy markets. In the US, electric power generation accounts for 30% of the natural gas demand (over 50% in the summer), 90% of the coal demand, and over 45% of the residual fuel oil demand [19]. Moreover, in the US natural gas market, for the past four years, the demand from electric power generation has been significantly and steadily increasing while demands from all the other sectors (industrial, commercial, and residential) have been slightly decreasing [20]. For example, in New England (the northeastern region of the US consisting of the states of Connecticut, Massachusetts, Rhode Island, Vermont, New Hampshire, and Maine), since 1999, approximately 97 percent of all the newly installed generating capacity has depended partially or entirely on natural gas [33]. Hence, various energy markets are inevitably and constantly interacting with electric power supply chains. For instance, from December 1, 2005 to April 1, 2006, the wholesale electricity price in New England decreased by 38% mainly because the delivered natural gas price declined by 45% within the same period. For another example, in August, 2006, the natural gas price jumped 14% because hot weather across the US led to elevated demand for electricity. This high electricity demand also caused the crude oil price to rise by 1.6% [29]. Similarly, the natural gas future price for September 2007 increased by 4.7% mainly because of the forecasted high electricity demands in Northeastern and Mid-western cities due to rising temperatures [62]. However, the quantitative connections between electric power supply chains and other fuel markets are not straightforward and depend on many factors, such as, the generating unit portfolios of power generating companies or generators (gencos), the technological characteristics of generating units as well as the the underlying physical transportation/transmission networks (with their associated capacities). Moreover, the availability and the reliability of diversified fuel supplies affect not only economic efficiency but also national security. For example, in January 2004, over 7000MW (megawatts or one million watts) of electric power generation, which accounts for almost one fourth of the total capacity of New England, was unavailable during the electric system peak due to the limited natural gas supply [37]. For another instance, the American Associ- 3

4 ation of Railroads has requested that the Federal Energy Regulatory Commission (FERC) investigate the reliability of the energy supply chain with a focus on electric power and coal transportation [6]. The relationships between electric power supply chains and other energy markets have drawn considerable attention from researchers in various fields. Emery and Liu [16] empirically estimated the cointegration of electric power futures and natural gas futures. Routledge, Seppi, and Spatt [63] focused on the connections between natural gas and electricity markets, and studied the equilibrium pricing of electricity contracts (see also [5]). Deng, Johnson, and Sogomonian [14] applied real option theory to develop models that utilize the relationship between fuel prices and electricity prices to value electricity generation and transmission assets. Huntington and Schuler [34] pointed out that the natural gas price was influenced by the residual fuel oil price because of the responsiveness of dual-fuel generating units in electric power networks (see also [2, 7]). The interesting interactions among oil, electric power, and natural gas markets will also be quantitatively investigated and generalized using our theoretical model and empirical examples. We note that Matsypura, Nagurney, and Liu [47] proposed the first network model that integrated fuel supplier networks and electric power supply chain networks. However, their model focused on the transactions of the electricity ownership and the economic decision-making processes of the market participants, and did not consider the physical constraints in electricity transmission networks and the electricity demand variations. Furthermore, no empirical results were presented. In this paper, we develop a new electric power network model with fuel supply markets that considers both the economic network transactions in energy supply chains and the physical network transmission constraints in the electric power network which are critical to the understanding of the regional differences in electric power prices [9, 10, 30]. We then apply the model to a specific case, the New England electric power supply chain. In particular, we present an empirical case study that demonstrates that the regional electric power prices simulated by our model match the actual electricity prices in New England very well. We also compute the electric power prices under natural gas and oil price variations. The empirical examples illustrate that both the generating unit responsiveness and the electric power market responsiveness are crucial to the full understanding and estimation of the impact of the residual fuel oil price on the natural gas price. Finally, we utilize the model to quantita- 4

5 tively investigate how changes in the demand for electricity influence the electric power and the fuel markets. This paper is organized as follows. In Section 2, we propose the integrated electric power supply chain and fuel supply network model. In Section 3, we provide some qualitative properties of the model and in Section 4, we discuss the computation of solutions to the model. In Section 5, we present a case study of the model applied to the New England electric power supply chain network. The empirical examples demonstrate how the theoretical model can be applied to investigate the interactions between electric power supply chains and fuel supply markets. The empirical application consists of 6 states, 5 fuel types, 82 power generators, with a total of 573 generating unit and generator combinations, and 10 demand market regions. Section 6 summarizes and concludes the paper and presents suggestions for future research. 2. The Integrated Electric Power Supply Chain and Fuel Supply Market Network Model In this section, we develop the electric power supply chain network model which includes regional electricity markets and fuel supply markets. 2.1 A Brief Introduction The electric power supply chain network model proposed in this paper includes four major components: the fuel supply markets, the power generators, the power buyers at demand markets, and the independent system operator (ISO). The power generators or gencos purchase fuels from the supply markets and produce electric power at the generating units. Gencos can sell electric power to the power buyers at the demand markets directly through bilateral contracts or they can sell to the power pool which is managed by the ISO. Additionally, gencos can also sell their capacities in the regional operating reserve markets. We assume that each electric power generator seeks to determine the optimal production and allocation of the electric power in order to maximize his own profit. 5

6 The power buyers at the demand markets search for the lowest electricity price. They can purchase electric power either directly from the gencos or from the power pool. For example, in the New England electric power supply chain, about 75 80% of electricity is traded through bilateral contracts, while about 20 25% of electricity is traded through the power pool [43]. Thus, the power pools function as markets that balance the residuals of supply and demand, and clear the regional wholesale electricity markets [43]. In most deregulated electricity markets, there is a non-profit independent system operator (ISO) who manages the wholesale electricity market, maintains system reliability, and oversees the transmission network to ensure system security (e.g. New England ISO, PJM Interconnection, and Electric Reliability Council of Texas, One of the ISO s objectives is to minimize the total cost and to achieve economic efficiency of the system by maintaining a competitive wholesale electricity market. The ISO undertakes two critical responsibilities in order to achieve a competitive and efficient wholesale market. First, the ISO dispatches the generating units that participate in the power pool, starting with the generating unit that offers the lowest supply price. In each regional power pool, all power suppliers will be paid at the same unit price which is equal to the market clearance price in that region. Secondly, the ISO schedules all transmission requests and monitors the entire transmission network; he also charges the network users congestion fees if certain transmission interface limits are reached [9, 10, 30]. Moreover, the ISO manages the operating reserve markets where the power generators can get paid for holding back their capacities to help to ensure system reliability. These major features of the ISO are fully reflected in our model. We let 1,..., a,..., A denote the types of fuels. We assume that there are M supply markets for each type of fuel. We assume that the electric power supply chain network includes 1,..., r,..., R regions which can be defined based on electricity transmission network interfaces. We let 1,..., g,..., G denote the gencos who may own and operate multiple generating units which may use different generating technologies and are located in various regions. For example, the genco, Con Edison Energy, owns one generating unit in New Hampshire, one generating unit in southeastern Massachusetts, and six generating units in western and central Massachusetts [39]. In particular, we let N gr denote the number of generating units owned by genco g in region r. Note that in each region r, different gencos may 6

7 have a different number of generating units. If genco g has no generating unit in region r, then N gr = 0. We let N G Rr=1 g=1 N gr denote the total number of generating units in the network. We assume that in each region there exist K demand market sectors which can be distinguished from one another by the types of associated consumers and the electricity consumption patterns. The top-tiered nodes in the electric power supply chain network in Figure 1 represent the AM fuel supply markets. The nodes at the second tier in Figure 1 represent the generating units associated with the gencos and regions. The three indices of a generating unit indicate the genco that owns the unit, the region of the unit, and the sequential identifier of the unit, respectively. For example, in region 1, the node on the left denotes genco 1 s first generating unit in region 1 while the node on the right denotes genco G s N th G1 (last) generating unit in region 1. The bottom-tiered nodes in Figure 1 represent the RK region and demand market combinations. Our model focuses on a single period, the length of which can be from several days to a month. We assume that the prices charged by the fuel suppliers in transacting with the gencos are relatively stable and do not change throughout the period. However, the electric power demands and prices may exhibit a strong periodic pattern within a day with the highest price typically being two to four times higher than the lowest price. Hence, we allow the electric power prices and demands to vary frequently within the study period. Note that there are two typical ways to represent electric power demand variations: load curves and load duration curves. A load curve plots electricity demands in temporal sequence while a load duration curve sorts and plots demand data according to the magnitude of the demands (e.g. [27, 46]). A point on the load duration curve represents the proportion of time that the demand is above certain level. In this paper, we use the discretized load duration curve to represent the demand variations within a period. In particular, we divide the load duration curve of the study period into 1,..., w,..., W blocks with L w denoting the time length of block w. In our empirical examples, using the New England electric power supply chain network as a case study in Section 5, we assume that there are six demand levels, consisting of two peak demand levels, two intermediate demand levels, and two low demand levels; hence, W = 6. 7

8 Fuel Markets for Fuel Type 1 Fuel Markets for Fuel Type a Fuel Markets for Fuel Type A 1, 1 1, M a, 1 a, M A, 1 A, M Generating Units of Gencos in Regions (genco, region, unit) 1, 1, 1 G, 1, N G1 1, r, 1 G, r, N Gr 1, R, 1 G, R, N GR Power Pool 1, 1 1, K r, 1 r, K R, 1 R, K Demand Market Sectors Demand Market Sectors Demand Market Sectors Region 1 Region r Region R Figure 1: The Electric Power Supply Chain Network with Fuel Supply Markets 8

9 We now summarize the critical assumptions of the model: 1. The model focuses on a single period, the length of which can be from several days to a month. 2. The fuel prices are relatively stable and do not fluctuate within the study period while the regional electricity prices fluctuate significantly as the demands vary within the study period. 3. The regions can be defined based on the transmission network interfaces. 4. Each genco can own and operate multiple generating units which may use different technologies and are located in different regions. A genco is an economic entity and is not restricted to a specific region while a genco s generating units are physically related to regions. 5. Each genco maximizes his own profit. 6. The decisions made by each genco include the quantities of fuels purchased, the production level of each of his generating units at each demand level, the sales of electricity to the power buyers and to the power pool at each demand level, and the amount of capacity sold at the operating reserve market. 7. Power buyers search for the lowest electric power price. 8. Power buyers can purchase electricity directly from the gencos through bilaterial contracts or from the power pool. 9. The ISO is a non-profit organization that maintains a competitive wholesale electricity market and ensures system security and reliability. 10. The ISO dispatches the generating units that participate in the power pool, starting with the generating unit that offers the lowest supply price. 11. The ISO schedules all transmission requests, monitors the transmission network, and charges congestion fees if certain transmission interface limits are reached. 12. The ISO manages operating reserve market to ensure system reliability. 9

10 2.2 The Model of the Integrated Electric Power Supply Chain Network with Fuel Supply Markets We now present the electric power supply chain network model. We first describe the behavior of the fuel suppliers, the gencos, the ISO, and the power buyers at demand markets. We then state the equilibrium conditions for the electric power supply chain network and provide the variational inequality formulation. Before we introduce the model, we would like to explain the use of r 1 and r 2 in the notation of the model. In general, a transaction of electricity is related to two regions (nodes): the region of injection where the generating unit is located and the region of withdrawal where the power buyer is located. In order to avoid confusion of the two regions, we use r 1 and r 2 to present the region of injection and the region of withdrawal, respectively. In our notation system, the m th supply market of fuel a is denoted by am; the u th generating unit owned by genco g in region r 1 is denoted by gr 1 u; and the k th demand market sector in region r 2 is denoted by r 2 k. For succinctness, we give the notation for the model in Tables 1, 2, 3, and 4. An equilibrium solution is denoted by. All vectors are assumed to be column vectors, except where noted. If a variable or cost function is related to the transaction of fuels, the superscript indicates the fuel market while the subscript indicates the generating unit and the demand level; if a variable or cost function is related to the bilateral transaction of electric power, the superscript indicates the generating unit while the subscript indicates the demand market and the demand level; and if a variable or cost function is related to electricity production or operating reserve, there is no superscript while the subscript indicates the generating unit as well as the demand level. The actual units for the prices, the demands, and the electric power flow transactions used in practice are explicated in Section 5, which contains the empirical case study and the associated examples. 10

11 Table 1: Parameters in the Electric Power Supply Chain Network Model Notation Definition q am The aggregated demand from sectors except that of electric power generation at the m th supply market of fuel a. c am gr 1 uw The unit transportation/transaction cost between the m th supply market of fuel a and the u th generating unit of genco g in region r 1 at demand level w. L w The time duration of demand level w. T Cap b The interface (flowgate) limit of interface b. Cap gr1 u The generating capacity of the u th generating unit of genco g in region r 1. OP gr1 u The maximum level of operating reserve that can be provided by the u th generating unit of genco g in region r 1. OP R r1 w The operating reserve requirement of region r 1 at demand level w. β gr1 ua (a 0) The nonnegative conversion rate (the inverse of the heat rate) at the u th generating unit of genco g in region r 1 if the generating unit utilizes fuel a; β gr1 ua is equal to zero if the generating unit does not use fuel a. We assume that for units using renewable technologies, all β gr1 uas are equal to zero. β gr1 u0 The renewable unit indicator. β gr1 u0 is equal to one if the u th generating unit of genco g in region r 1 utilizes renewable technologies and zero otherwise. For generating units using renewable technologies, all β gr1 uas are equal to zero. α r1 r 2 b The impact of transferring one unit electricity from region r 1 to region r 2 on interface b. α r1 r 2 b is equal to P T DF r2 b P T DF r1 b where P T DF rb denotes the power transmission distribution factor of region (node) r for interface limit b. In particular, P T DF rb is defined as the quantity of power flow (MW) through the critical link of interface b induced by a 1 MW injection at node r [9, 10, 30]. d r2 kw The fixed demand at demand market sector k in region r 2 at demand level w. κ r2 w The transmission loss factor of region r 2 at demand level w. 11

12 Table 2: Decision Variables in the Electric Power Supply Chain Network Model Notation h Q 1 q w Q 2 w Yw 1 Yw 2 Z w Definition AM-dimensional vector of fuel supplies at the fuel markets with component h am denoting the total supply of fuel a at the m th supply market of fuel a. AMNW -dimensional vector of fuel flows between fuel supply markets and generating units within the entire period with component magr 1 uw denoted by qgr am 1 uw and denoting the transactions between the m th supply market of fuel a and the u th generating unit of genco g in region r 1 at demand level w. N-dimensional vector of the power generators electric power outputs at demand level w with components q gr1 uw denoting the power generation at the u th generating unit of genco g in region r 1 at demand level w. We group the q w at all demand levels w into vector q. NRK-dimensional vector of electric power flows between generating units and demand markets at demand level w with component gr 1 ur 2 k denoted by q gr 1u r 2 kw and denoting the transactions between the u th generating unit of genco g in region r 1 and demand market sector k in region r 2 at demand level w. We group the Q 2 w at all demand levels w into vector Q 2. NR-dimensional vector of electric power transactions between power generators and regional power pools at demand level w with component gr 1 ur 2 denoted by y gr 1u r 2 w and denoting the transactions between generating unit u of generator g in region r 1 and the regional power pool in region r 2 at demand level w. We group the Yw 1 at all demand levels w into vector Y 1. R 2 K-dimensional vector of electric power transactions between demand markets and regional power pools at demand level w with component r 1 r 2 k denoted by y r 1 r 2 kw and denoting the transactions between demand market sector k in region r 2 and the regional power pool in region r 1 at demand level w. We group the Yw 2 at all demand levels w into vector Y 2. N-dimensional vector of regional operating reserves with component gr 1 u denoted by z gr1 uw and denoting the operating reserve held by generating unit u of genco g in region r 1 at demand level w. We group the Z w at all demand levels w into vector Z. 12

13 Table 3: Endogenous Prices and Shadow Prices of the Electric Power Supply Chain Network Model Notation Definition ρ am gr 1 uw The price offered by generating unit u of genco g in region r 1 at demand level w in transacting with the m th supply market of fuel a. We group all ρ am gr 1 uws into vector ρ 1. ρ gr 1u r 2 kw The unit price charged by generating unit u of genco g in region r 1 for the transaction with power buyers in demand market sector k in region r 2 at demand level w. We group all ρ gr 1u r 2 kw s into vector ρ 2. ρ r2 w The unit electricity price at location r 2 on the electricity pool market at demand level w. We group all ρ r2 ws into vector ρ 3. ρ r2 kw The unit electric power price offered by the buyers at demand market sector k in region r 2 at demand level w. We group all ρ r2 kws into vector ρ 4. ϕ r1 w The unit price of capacity on the regional operating reserve market in region r 1 at demand level w. We group all ϕ r1 ws into vector ϕ. µ bw The unit congestion charge of interface (flowgate) b at demand level w. We group all µ bw s into vector µ. η gr1 uw The shadow price associated with the capacity constraint of generating unit u of generator g in region r 1 at demand level w. We group all η gr1 uws into vector η. λ gr1 uw The shadow price associated with the maximum operation reserve constraint of generating unit u of generator g in region r 1 at demand level w. We group all λ gr1 uws into vector λ. 13

14 Table 4: Cost and Price Functions of the Electric Power Supply Chain Network Model Notation Definition π am (h) The inverse supply function (price function) at the m th supply market of fuel a. f gr1 uw(q gr1 uw) The generating cost of generating unit u of genco g in region r 1 at demand level w. c gr 1u r 2 kw (qgr 1u r 2 kw ) The transaction/transmission cost incurred at generating unit u of genco g in region r 1 in transacting with demand market sector k in region r 2 at demand level w. c gr 1u r 2 w (y gr 1u r 2 w ) The transaction/transmission cost incurred at generating unit u of genco g in region r 1 in selling electricity to region r 2 through the power pool at demand level w. c gr1 uw(z gr1 uw) The operating reserve cost at generating unit u of genco g in region r 1 at demand ĉ gr 1u r 2 kw (Q2 w) ĉ r 1 r 2 kw (Y 2 w) level w. The unit transaction/transmission cost incurred by power buyers in demand market sector k in region r 2 in transacting with generating unit u of genco g in region r 1 at demand level w. The unit transaction/transmission cost incurred by power buyers in demand market sector k in region r 2 when purchasing electricity from region r 1 through the power pool at demand level w. The Equilibrium Conditions for the Fuel Supply Markets We first describe the equilibrium conditions for the fuel supply markets. The typical fuels used for electric power generation include coal, natural gas, residual fuel oil (RFO), distillate fuel oil (DFO), jet fuel, and uranium. We assume that the fuel suppliers take into account the prices offered by the power generators and the transportation/distribution costs in making their economic decisions (see also [24, 25, 64]). We assume that the following conservation of flow equations must hold for all fuel supply markets a = 1,..., A; m = 1,..., M: W G R N gr1 w=1 g=1 r 1 =1 u=1 q am gr 1 uw + q am = h am, (1) where parameter q am denotes the aggregated demand from the other sectors. Equation (1) states that the total supply of fuel a at market am is equal to the sum of the demand from electric power generation and the aggregated demand from the other sectors. 14

15 We assume that the inverse supply function (price function), π am (h), is known for each fuel supply market am. We assume that the fuel price in a market depends not only on the supply of fuels in that market but also on the supplies of fuels at the other fuel markets. We also assume that π am (h) is a non-decreasing continuous function. A special case is where π am (h) = π am + 0 h = π am in which case the fuel price is fixed and equal to π am. The (spatial price) equilibrium conditions (cf. [49]) for suppliers at fuel supply market am; a = 1,..., A; m = 1,..., M, take the form: for each generating unit gr 1 u; g = 1,..., G; r 1 = 1,..., R; u = 1,..., N gr1, and at each demand level w: equivalently, in view of (1), π am (h ) + c am gr 1 uw π am (Q 1 ) + c am gr 1 uw { = ρ am gr 1 uw, if q am ρ am gr 1 uw, if q am { = ρ am gr 1 uw, if q am ρ am gr 1 uw, if q am gr 1 uw > 0, gr 1 uw = 0; gr 1 uw > 0, gr 1 uw = 0. (2) (3) In equilibrium, conditions (3) must hold simultaneously for all the fuel supply market and generating unit pairs and at all demand levels. We can express these equilibrium conditions as the following variational inequality (see, e.g., [49]): determine Q 1 K 1, such that W A M G R N gr1 w=1 a=1 m=1 g=1 r 1 =1 u=1 where K 1 {Q 1 Q 1 R AMNW + }. [ ] πam (Q 1 ) + c am gr 1 uw ρ am gr 1 uw [q am gr1 uw qgr am 1 uw] 0, Q 1 K 1, (4) The Behavior of the Power Generators and Their Optimality Conditions Recall that the equilibrium prices are indicated by *. Under the assumption that each individual genco is a profit-maximizer and may own multiple generating units in various regions, the optimization problem of genco g can be expressed as follows: Maximize W R N gr1 R K L w ρ gr 1u r 2 kw qgr 1u r 2 kw w=1 r 1 =1 u=1 r 2 =1 k=1 W R N gr1 R + L w ρ r 2 wy gr 1u r 2 w + w=1 r 1 =1 u=1 r 2 =1 W R N gr1 w=1 r 1 =1 u=1 L w ϕ r 1 wz gr1 uw 15 W A M R N gr1 w=1 a=1 m=1 r 1 =1 u=1 ρ am gr 1 uwq am gr 1 uw

16 W R N gr1 L w w=1 r 1 =1 u=1 W R N gr1 R K f gr1 uw(q gr1 uw) L w w=1 r 1 =1 u=1 r 2 =1 k=1 c gr 1u r 2 kw (qgr 1u r 2 kw ) W R N gr1 R W R N L w c gr 1u r 2 w (y gr gr1 1u r 2 w ) L w c gr1 uw(z gr1 uw) w=1 r 1 =1 u=1 r 2 =1 w=1 r 1 =1 u=1 subject to: W R N gr1 R B K L w µ bwα r1 r 2 b[ q gr 1u r 2 kw + ygr 1u r 2 w ] (5) w=1 r 1 =1 u=1 r 2 =1 b=1 k=1 R K r 2 =1 k=1 q gr 1u r 2 kw + R r 2 =1 y gr 1u r 2 w = q gr1 uw, r 1 = 1,..., R; u = 1,..., N gr1 ; w = 1,..., W, (6) A M β gr1 ua qgr am 1 uw+l w β gr1 u0q gr1 uw = L w q gr1 uw, r 1 = 1,..., R; u = 1,..., N gr1 ; w = 1,..., W, a=1 m=1 (7) q gr1 uw + z gr1 uw Cap gr1 u, r 1 = 1,..., R; u = 1,..., N gr1 ; w = 1,..., W, (8) z gr1 uw OP gr1 u, r 1 = 1,..., R; u = 1,..., N gr1 ; w = 1,..., W, (9) q gr 1u r 2 kw 0, r 1 = 1,..., R; u = 1,..., N gr1 ; r 2 = 1,..., R; k = 1,..., K; w = 1,..., W, (10) q am gr 1 uw 0, a = 1,..., A; m = 1,..., M; r 1 = 1,..., R; u = 1,..., N gr1 ; w = 1,..., W, (11) y gr 1u r 2 w 0, r 1 = 1,..., R; u = 1,..., N gr1 ; r 2 = 1,..., R; w = 1,.., W ;, (12) z gr1 uw 0, r 1 = 1,..., R; u = N gr1 ; w = 1,..., W. (13) The first three terms in the objective function represent the revenues from bilateral transactions with the demand markets, the energy pool sales, and the regional operating reserve market, respectively. The fourth term is the total payout to the fuel suppliers. The fifth, sixth, and seventh terms represent the generating cost, the transaction costs of bilateral contracts with the demand markets, and the transaction costs of selling electric power to energy pools, respectively. The eighth term is the cost of providing operating reserve z gr1 uw. The last term of the objective function represents the cost of congestion charges where B b=1 µ bwα r1 r 2 b is equivalent to the congestion charge of transferring one unit electricity from region r 1 to region r 2. Note that α r1 r 2 b is equal to P T DF r2 b P T DF r1 b where P T DF rb denotes the power transmission distribution factor of region (node) r for interface limit b. In particular, 16

17 P T DF rb is defined as the quantity of power flow (MW) through the critical link of interface b induced by a 1 MW injection at node r [9, 10, 30]. Here, we assume that the gencos have to pay the transmission right costs for bilateral transactions. Next, we explain the constraints genco g must satisfy when he maximizes the profit. Constraint (6) states that at each generating unit the total amount of electric power sold cannot exceed the total production of electric power. Constraint (7) models the production of electricity at each generating unit. If a generating unit uses fossil fuel, at each demand level, the quantity of electricity produced is equal to the quantity of electricity converted from the fuels. In constraint (7), β gr1 ua is equal to the nonnegative conversion rate (the inverse of the heat rate) at generating generating unit u of genco g in region r 1 if the unit utilizes fuel a, and is equal to zero otherwise; β gr1 u0 is equal to one if the generating unit utilizes renewable technologies and is zero otherwise. We assume that for units using renewable technologies, all β gr1 uas are equal to zero. Note that for renewable generating units, constraint (7) will automatically hold. In the electric power industry, generating units that burn the same type of fuel may have very different average heat rates depending on the technologies that the generating units use. For example, the heat rates of large natural gas generating units range from 5500 Btu/kWh to Btu/kWh while the heat rates of large oil generating units vary from 6000 Btu/kWh to Btu/kWh (e.g. [67]). Constraint (8) states that the sum of electric power generation and operating reserve cannot exceed generating unit capacity, Cap gr1 u. Constraint (9), in turn, states that the operating reserve provided by generating unit u of genco g in region r 1 cannot exceed the maximum level of operating reserve of that unit, OP gr1 u. We assume that the generating cost and the transaction cost functions for the generating units are continuously differentiable and convex, and that the gencos compete in a noncooperative manner in the sense of Cournot [13] and Nash [59, 60]. The optimality conditions for all power generators simultaneously, under the above assumptions (see also [3, 4, 23, 49]), coincide with the solution of the following variational inequality: determine 17

18 (Q 1, q, Q 2, Y 1, Z, η, λ ) K 2 satisfying R gr1 [ fgr1 uw(q ] gr L 1 uw) w + ηgr w=1 g=1 r 1 =1 u=1 q 1 uw [q gr1 uw qgr 1 uw] gr1 uw R gr1 [ R K c gr 1 u r + L 2 kw (qgr 1u r 2 kw ) w w=1 g=1 r 1 =1 u=1 r 2 =1 k=1 q gr 1u + r 2 kw R gr1 [ R c gr 1 u r + L 2 w (y gr 1u r 2 w ) B w w=1 g=1 r 1 =1 u=1 r 2 =1 y gr 1u + r 2 w B b=1 b=1 ] µ bwα r1 r 2 b ρ gr 1u r 2 kw [q gr 1u r 2 kw qgr 1u r 2 kw ] µ bwα r1 r 2 b ρ r 2 w ] [y gr 1u r 2 w y gr 1u r 2 w ] N W G R gr1 A M + ρ am gr 1 uw [qgr am 1 uw qgr am 1 uw] w=1 g=1 r 1 =1 u=1 a=1 m=1 R gr1 [ cgr1 uw(z ] gr + L 1 uw) w + λ gr w=1 g=1 r 1 =1 u=1 z 1 uw + ηgr 1 uw ϕ r 1 w [z gr1 uw zgr 1 uw] gr1 uw R gr1 [ ] + L w Capgr1 u qgr 1 uw zgr 1 uw [ηgr1 uw ηgr 1 uw] w=1 g=1 r 1 =1 u=1 R gr1 [ ] + L w OPgr1 u zgr 1 uw [λgr1 uw λ gr 1 uw] 0, (Q 1, q, Q 2, Y 1, Z, η, λ) K 2, w=1 g=1 r 1 =1 u=1 (14) where K 2 {(Q 1, q, Q 2, Y 1, Z, η, λ) (Q 1, q, Q 2, Y 1 AMNW +NRKW +NRW +4NW, Z, η, λ) R+, and (6) and (7) hold}. The ISO s Role The ISO manages the wholesale electricity market, maintains system reliability, and oversees the transmission network to ensure system security. The ISO dispatches the power generators that participate in the power pool, starting with the genco that offers the lowest supply price. In each regional power pool, all power suppliers will receive the same unit price which is equal to the market clearance price in that region. The ISO achieves this economic efficiency by providing and overseeing competitive power pools as well as maintaining market clearance at each power pool. In the previous section, we have assumed that at power pools the gencos compete with one another in a 18

19 noncooperative manner in the sense of Cournot [13] and Nash [59, 60], and have incorporated this competition in (14). The competition among the generators and the maintenance of market clearance in the regional power pools lead to economic dispatches of electric power generation. The ISO ensures that the regional electricity markets r = 1,..., R clear at each demand level w = 1,..., W, that is, equivalently, G R N gr1 g=1 r 1 =1 u=1 G N R gr1 L w g=1 r 1 =1 u=1 y gr 1u rw y gr 1u rw { = Rr2 Kk=1 =1 y r R Kk=1 r2 =1 y r { = Rr2 Kk=1 Lw =1 y r L Rr2 Kk=1 w =1 y r r 2 kw, if ρ rw > 0, r 2 kw, if ρ rw = 0; r 2 kw, if ρ rw > 0, r 2 kw, if ρ rw = 0. (15) (15a) Here, the left-hand side of constraint (15) (or (15a)) represents the total quantity of electric power sold by power sellers at region r through the power pool, and the right-hand side represents the total amount of electric power purchased by power buyers from region r through the power pool. We would like to note that (14) and (15) (or (15a)) together ensure that in equilibrium, at demand level w, the generating units that participate in the power pool in region r and have lower marginal costs than ρ rw will be paid at the same unit price which is equal to the market clearance price ρ rw. Moreover, the ISO also manages operating reserve markets where the gencos can get paid for holding back their capacities to help to ensure system reliability. We have assumed that the generators compete in the operating reserve markets in a noncooperative manner. The ISO needs to ensure that the regional operating reserve markets r 1 = 1,..., R clear at each demand level w = 1,..., W, that is, equivalently, N G gr1 zgr 1 uw g=1 u=1 G N gr1 L w zgr 1 uw g=1 u=1 { = OP Rr1 w, if ϕ r 1 w > 0, OP R r1 w, if ϕ r 1 w = 0; { = Lw OP R r1 w, if ϕ r 1 w > 0, L w OP R r1 w, if ϕ r 1 w = (16) (16a)

20 The ISO also manages transmission congestion and impose congestion fees, which not only ensures system security, but also eliminates inter-region arbitrage opportunities and enhance economic efficiency of the system. We use a linearized direct current network to approximate the transmission network, and assume that the ISO charges network users congestion fees in order to ensure that the interfaces limits are not violated [9, 10, 30]. In our model, the following conditions must hold for each interface b and at each demand level w, where b = 1,..., B; w = 1,..., W : N R R G gr1 K [ q gr 1u G N r 2 kw + gr1 r 1 =1 r 2 =1 g=1 u=1 k=1 g=1 u=1 y gr 1u r 2 w + K k=1 y r 1 r 2 kw ]α r 1 r 2 b { = T Capb, if µ bw > 0, T Cap b, if µ bw = 0; (17) equivalently, R N R G gr1 K L w [ q gr 1u r 2 kw r 1 =1 r 2 =1 g=1 u=1 k=1 + G N gr1 g=1 u=1 y gr 1u r 2 w + K k=1 y r 1 r 2 kw ]α r 1 r 2 b { = Lw T Cap b, if µ bw > 0, L w T Cap b, if µ bw = 0. (17a) In equilibrium, conditions (15), (16), and (17) (equivalently, (15a), (16a) and (17a)) must hold simultaneously. We can express these equilibrium conditions using the following variational inequality: determine (µ, ρ 3, ϕ ) R W B+2W R +, such that W B N R R G gr1 K L w [T Cap b [ q gr 1u r 2 kw w=1 b=1 r 1 =1 r 2 =1 g=1 u=1 k=1 W R N G R gr1 + L w [ w=1 r=1 g=1 r 1 =1 u=1 y gr 1u rw + G N gr1 g=1 u=1 R r 2 =1 k=1 y gr 1u r 2 w + K K k=1 y r r 2 kw] [ρ rw ρ rw] y r 1 r 2 kw ]α r 1 r 2 b] [µ bw µ bw] W R N G gr1 + L w [ zgr 1 uw OP R r1 w] [ϕ r1 w ϕ r 1 w] 0, w=1 r 1 =1 g=1 u=1 BW +2RW (µ, ρ 3, ϕ) R+. (18) 20

21 Equilibrium Conditions for the Demand Markets Next, we describe the equilibrium conditions at the demand markets. We assume that the consumers in the same demand market sector are homogenous and have the same consumption pattern. The consumers search for the lowest electricity cost which is equal to the sum of the electricity price and the transaction cost. We assume that all demand market sectors in all regions have fixed and known demands, and the following conservation of flow equations, hence, must hold for all regions r 2 = 1,..., R, all demand market sectors k = 1,..., K, and at all demand levels w = 1,..., W : G R N gr1 g=1 r 1 =1 u=1 q gr 1u r 2 kw + R r 1 =1 y r 1 r 2 kw = (1 + κ r 2 w)d r2 kw, (19) where d r2 kw denotes the demand at market sector k in region r 2 at demand level w, and κ r2 w denotes the transmission loss factor which is usually between 1% 6%. We also assume that all unit transaction cost functions ĉ gr 1u r 2 kw (Q2 w)s and ĉ r 1 r 2 kw (Y 2 w)s are continuous and nondecreasing. The equilibrium conditions for consumers at demand market sector k in region r 2 take the form: for each generating unit gr 1 u; g = 1,..., G; r 1 = 1,..., R; u = 1,..., N gr1, and each demand level w; w = 1,..., W : { ρ gr 1u r 2 kw + ĉgr 1u = ρ r 2 kw (Q2 w ) r2 kw, if q gr 1u r 2 kw > 0, ρ r 2 kw, if q gr 1u r 2 kw = 0; (20) equivalently, and equivalently, { L w [ρ gr 1u r 2 kw + ĉgr 1u = Lw ρ r 2 kw (Q2 w )] r 2 kw, if q gr 1u r 2 kw > 0, L w ρ r 2 kw, if q gr 1u r 2 kw = 0; (20a) ρ r 1 w + B b=1 { µ bwα r1 r 2 b + ĉ r 1 2 = ρ r 2 kw (Yw ) r2 kw, if y r 1 r 2 kw > 0, ρ r 2 kw, if y r 1 r 2 kw = 0; (21) L w [ρ r 1 w + B b=1 { µ bwα r1 r 2 b + ĉ r 1 2 = Lw ρ r 2 kw (Yw )] r 2 kw, if y r 1 r 2 kw > 0, L w ρ r 2 kw, if y r 1 r 2 kw = 0. (21a) 21

22 Conditions (20) and (20a) state that, in equilibrium, if power buyers at demand market sector k in region r 2 purchase electricity from generating unit u of genco g in region r 1, then the price the consumers pay is exactly equal to the sum of the electricity price and the unit transaction cost. However, if the electricity price plus the transaction cost is greater than the price the buyers are willing to pay at the demand market, there will be no transaction between this generating unit / demand market pair. Conditions (21) and (21a) state that power buyers in demand markets need to also consider congestion fees when they purchase electric power from other regions through the power pool. In equilibrium, conditions (20) and (21) (equivalently, (20a) and (21a)) must hold simultaneously for all demand markets in all regions. We can express these equilibrium conditions using the following variational inequality: determine (Q 2, Y 2 ) K 3, such that R gr1 R K [ L w ρ gr 1 u r 2 kw + ĉgr 1u r 2 kw (Q2 w ) ] [q gr 1u r 2 kw qgr 1u r 2 kw ] w=1 g=1 r 1 =1 u=1 r 2 =1 k=1 W R [ ] R K B + L w ρ r 2 w + µ bwα r1 r 2 b + ĉ r 1 2 r 2 kw (Yw ) w=1 r 1 =1 r 2 =1 k=1 b=1 [y r 1 r 2 kw yr 1 r 2 kw ] 0, (Q 2, Y 2 ) K 3, (22) where K 3 {(Q 2, Y 2 ) (Q 2, Y 2 ) R NRKW +R2 KW + and (19) holds}. The Equilibrium Conditions for the Electric Power Supply Chain Network In equilibrium, the optimality conditions for all gencos, the equilibrium conditions for all fuel supply markets, all demand market sectors, and the independent system operator must be simultaneously satisfied so that no decision-maker has any incentive to alter his/her transactions. We now formally state the equilibrium conditions for the electric power supply chain with fuel supply markets as follows. Definition 1: Electric Power Supply Chain Network Equilibrium The equilibrium state of the electric power supply chain network with fuel supply markets is one where the fuel and electric power flows and the prices satisfy the sum of conditions (4), (14), (18), and (22). 22

23 We now state and prove: Theorem 1: Variational Inequality Formulation of the Electric Power Supply Chain Network Equilibrium Model with Fuel Suppliers The equilibrium conditions governing the electric power supply chain network according to Definition 1 coincide with the solution of the variational inequality given by: determine (Q 1, q, Q 2, Y 1, Y 2, Z, η, λ, µ, ρ 3, ϕ ) K 4 satisfying W A M G R N gr1 w=1 a=1 m=1 g=1 r 1 =1 u=1 [ ] πam (Q 1 ) + c am gr 1 uw [q am gr1 uw qgr am 1 uw] R gr1 [ fgr1 uw(q ] gr + L 1 uw) w + ηgr w=1 g=1 r 1 =1 u=1 q 1 uw [q gr1 uw qgr 1 uw] gr1 uw R gr1 [ R K c gr 1 u r + L 2 kw (qgr 1u r 2 kw ) ] B w w=1 g=1 r 1 =1 u=1 r 2 =1 k=1 q gr 1u + µ bwα r1 r 2 b + ĉ gr 1u r 2 kw (Q2 w ) [q gr 1u r 2 kw qgr 1u r 2 kw ] r 2 kw b=1 R gr1 [ R c gr 1 u r + L 2 w (y gr ] 1u r 2 w ) B w w=1 g=1 r 1 =1 u=1 r 2 =1 y gr 1u + µ bwα r1 r 2 b ρ r 2 w [y gr 1u r 2 w y gr 1u r 2 w ] r 2 w b=1 R gr1 [ cgr1 uw(z ] gr + L 1 uw) w + λ gr w=1 g=1 r 1 =1 u=1 z 1 uw + ηgr 1 uw ϕ r 1 w [z gr1 uw zgr 1 uw] gr1 uw W R [ ] R K B + L w ρ r 1 w + ĉ r 1 2 r 2 kw (Yw ) + µ bwα r1 r 2 b [y r 1 r 2 kw yr 1 r 2 kw ] w=1 r 1 =1 r 2 =1 k=1 b=1 R gr1 [ ] + L w Capgr1 u qgr 1 uw zgr 1 uw [ηgr1 uw ηgr 1 uw] w=1 g=1 r 1 =1 u=1 R gr1 [ ] + L w OPgr1 u zgr 1 uw [λgr1 uw λ gr 1 uw] w=1 g=1 r 1 =1 u=1 W B N R R G gr1 K + L w [T Cap b [ w=1 b=1 r 1 =1 r 2 =1 g=1 u=1 k=1 W R N G R gr1 + L w [ w=1 r=1 g=1 r 1 =1 u=1 q gr 1u r 2 kw + G y gr 1u rw N gr1 K y gr 1u r 2 w + g=1 u=1 k=1 R K r 2 =1 k=1 y r r 2 kw] [ρ rw ρ rw] y r 1 r 2 kw ]α r 1 r 2 b] [µ bw µ bw] 23

24 W + R N G gr1 L w [ w=1 r 1 =1 g=1 u=1 z gr 1 uw OP R r1 ] [ϕ r1 w ϕ r 1 w] 0, (Q 1, q, Q 2, Y 1, Y 2, Z, η, λ, µ, ρ 3, ϕ) K 4, (23) where K 4 {(Q 1, q, Q 2, Y 1, Y 2, Z, η, λ, µ, ρ 3, ϕ) (Q 1, q, Q 2, Y 1, Y 2, Z, η, λ, µ, ρ 3, ϕ) R AMNW +NRKW +NRW +4NW +R2 KW +BW +2RW + and (6), (7), and (19) hold}. Proof: We first establish that Definition 1 implies variational inequality (23). summation of (4), (14), (18), and (22), after algebraic simplifications, yields (23). Indeed, Now we prove the converse, that is, that a solution to (23) satisfies the sum of (4), (14), (18), and (22), and is, hence, an equilibrium. To variational inequality (23), add ρ am gr 1 uw ρ am gr 1 uw to the term in the first brackets preceding the first multiplication sign, and add ρ gr 1u r 2 kw ρgr 1u r 2 kw in the brackets preceding the third multiplication sign. The addition of such terms does not change (23) since the value of these terms is zero, and yields the sum of (4), (14), (18), and (22) after simple algebraic simplifications. Q.E.D. The variational inequality problem (23) can be rewritten in standard variational inequality form (cf. [49]) as follows: determine X K satisfying F (X ) T, X X 0, X K, (24) where X (Q 1, q, Q 2, Y 1, Z, Y 2, η, λ, µ, ρ 3, ϕ) T, K K 4, and F (X) (F am gr 1 uw, F gr 1u w, F gr 1u r 2 kw, F gr 1u r 2 w, F zgr 1u w, F r1 r 2 kw, F λgr1 u, F ηgr1 u, F bw, F rw, F r1 w) with indices a = 1,..., A; m = 1,..., M; w = 1,..., W ; r 1 = 1,..., R; r 2 = 1,..., R; r = 1,..., R; g = 1,..., G; u = 1,..., N gr1 ; k = 1,..., K; b = 1,..., B, and the functional terms preceding the multiplication signs in (23), respectively. Here <, > denotes the inner product in Ω-dimensional Euclidian space where Ω = AMNW +NRKW +NRW +4NW + R 2 KW + BW + 2RW. 24

25 3. Qualitative Properties In this section, we provide some qualitative properties of the solution to variational inequality (24); equivalently, (23). We can derive existence of a solution X to (24) simply from the assumption of continuity of functions that enter F (X), which is the case in this model [45, 49]. We now state the following theorems. Theorem 2: Existence If (Q 1, q, Q 2, Y 1, Y 2, Z, η, λ, µ, ρ 3, ϕ ) satisfies variational inequality (23) then (Q 1, q, Q 2, Y 1, Y 2, Z ) is a solution to the variational inequality problem: determine (Q 1, q, Q 2, Y 1, Y 2, Z ) K 5 satisfying W A M G R N gr1 w=1 a=1 m=1 g=1 r 1 =1 u=1 [ ] πam (Q 1 ) + c am gr 1 uw [q am gr1 uw qgr am 1 uw] R gr1 [ fgr1 uw(q ] gr + L 1 uw) w [q gr1 uw qgr w=1 g=1 r 1 =1 u=1 q 1 uw] gr1 uw R gr1 [ R K c gr 1 u r + L 2 kw (qgr 1u r 2 kw ) ] w w=1 g=1 r 1 =1 u=1 r 2 =1 k=1 q gr 1u + ĉ gr 1u r 2 kw (Q2 w ) [q gr 1u r 2 kw qgr 1u r 2 kw ] r 2 kw R gr1 R + L w w=1 g=1 r 1 =1 u=1 r 2 =1 c gr 1u r 2 w (y gr 1u r 2 w ) y gr 1u [y gr 1u r 2 w y gr 1u r 2 w ] r 2 w R gr1 c gr1 uw(zgr + L 1 uw) w [z gr1 uw zgr w=1 g=1 r 1 =1 u=1 z 1 uw] gr1 uw W R R K + L w ĉ r 1 2 r 2 kw (Yw ) [y r 1 r 2 kw yr 1 r 2 kw ] 0, (Q1, q, Q 2, Y 1, Y 2, Z) K 5, (25) w=1 r 1 =1 r 2 =1 k=1 where K 5 {(Q 1, q, Q 2, Y 1, Y 2, Z) (Q 1, q, Q 2, Y 1, Y 2, Z) R AMNW +NRKW +NRW +2NW +R2 KW + and (6), (7), (8), (9), (19), and G N R gr1 L w g=1 r 1 =1 u=1 G N gr1 L w g=1 u=1 y gr 1u rw L w R K yr r 2 kw, r; w, (26) r 2 =1 k=1 z gr1 uw L w OP R r1 w, r 1 ; w, (27) 25

26 R N R G gr1 K and L w [ r 1 =1 r 2 =1 g=1 u=1 k=1 are satisfied}. q gr 1u r 2 kw + G N gr1 K y gr 1u r 2 w + g=1 u=1 k=1 y r 1 r 2 kw ]α r 1 r 2 b L w T Cap b, b; w (28) A solution to (25) is guaranteed to exist provided that K 5 is nonempty. Moreover, if (Q 1, q, Q 2, Y 1, Y 2, Z ) is a solution to (25), there exist (η, λ, µ, ρ 3, ϕ 2NW +BW +2RW ) R+ with (Q 1, q, Q 2, Y 1, Y 2, Z, η, λ, µ, ρ 3, ϕ ) being a solution to variational inequality (23). Proof: The proof is an analog of the proof of Theorem 3 in [53]. Since all the constraints of K 5 are linear, it is easy to verify the existence of a feasible point in K 5. In our case study, because K 5 is nonempty for the New England electric power supply chain, the existence of a solution is guaranteed for each empirical example in Section 5. We now recall the concept of monotonicity and state an additional theorem. Theorem 3: Monotonicity Suppose that all cost functions in the model are continuously differentiable and convex; all unit cost functions are monotonically increasing, and the inverse price functions at the fuel supply markets are monotonically increasing. Then the vector F that enters the variational inequality (23) as expressed in (24) is monotone, that is, (F (X ) F (X )) T, X X 0, X, X K, X X. (29) Proof: See the Appendix. In our case study for New England, F (X) is monotone and the Jacobian of F (X) is uniformly positive semidefinite. Moreover, F (X) is linear and, hence, Lipschitz continuous (see [49]). 26